Continuity

A function is continuous at a point when three things line up: (1) f(a) is defined, (2) the limit as x → a exists, and (3) they are equal. Informally: you can draw through the point without lifting your pen.

Breaks come in flavours: a removable hole (fixable by plugging the gap), a jump (left and right disagree), and an infinite blow-up (a vertical asymptote, like 1/x at 0). Naming the break tells you how — or whether — it can be repaired.